What do emulsification failure and Bose - Einstein conden - sation have in common ?

– Ideal bosons and classical ring polymers formed via self-assembly, are known to have the same partition function, and so analogous phase transitions. In ring polymers, the analogue of Bose-Einstein condensation occurs when a ring polymer of macroscopic size appears. We show that a transition of the same general form occurs within a whole class of systems with self-assembly, and illustrate it with the emulsification failure of a microemulsion phase of water, oil and surfactant. As with Bose-Einstein condensation, the transition occurs even in the absence of interactions. Bose-Einstein condensation (BEC) is a textbook [1], but rather unusual, phase transition. It occurs for noninteracting bosons, in contrast to more conventional transitions such as that of the Ising model, which are driven by interactions. Feynman [2] showed that the statistical mechanics of bosons can be performed via what he called path integrals. These integrals are, in turn, equivalent to integrals over the configurations of ring polymers. Thus, the path integral formalism of Feynman implies that noninteracting bosons and self-assembling, noninteracting ring polymers have partition functions of exactly the same form. Necessarily then, ring polymers must undergo a phase transition precisely analogous to BEC [3]. Here, we generalise this result to show that there is a class of self-assembling systems which undergo a phase transition analogous to BEC. This phase transition occurs in the absence of interactions between the aggregates formed by self-assembly. At BEC a condensate appears which is a macroscopic number of bosons in a single state. In the analogous transition in self-assembling systems, an aggregate of macroscopic size appears. For the example considered here, a microemulsion, the macroscopic aggregate is a bulk oil phase. Self-assembling systems are systems in which the particles are not immutable objects but are formed reversibly [4, 5, 6]. Typically we have water, surfactant and sometimes oil. The Typeset using EURO-TEX 2 EUROPHYSICS LETTERS surfactant molecules then spontaneously assemble into micelles, or coat and stabilise droplets of oil. Both the micelles and the droplets are what we term aggregates. For example, in a microemulsion, equilibrium is obtained when the oil is dispersed in water as oil droplets whose surfaces are coated with surfactant. These droplets are more stable than just a single bulk oil phase because with a single bulk phase there is no extensive oil-water interface, and the amphiphilic surfactant molecules have the lowest energy at this interface. The distribution of droplets changes with density and temperature. A BEC-like transition occurs when, at equilibrium, some of the oil exists as a macroscopic droplet, i.e., a bulk phase. This transition is called emulsification failure [7]. Let us begin by outlining BEC in Feynman’s language [2]. The partition function of N bosons is given in terms of their density matrix ρ̂ by ZN (β, V ) = trV ρ̂, ρ̂ ≡ e −βHN , (1) with HN the Hamiltonian operator, β = 1/kT . The density matrix of noninteracting identical bosons can be written in terms of 1-particle density matrices, ρ̂1 (